Schrodinger field

If we take the non-relativistic limit of either the K.G. field equation or the Dirac field equation(where the field obeys commutator relations),we get the Schrodinger equation.Obviously at no point does the field turn into a wavefunction---so the Schrodinger equation also is an equation for the quantum field.The field obeys microscopic causality i.e. the field at x1,t1 does not commute with the field at x2,t2(time-like separated from x1,t1).Now my question is:--if we take this into account in phenomena of the condensed matter or in many body theory,would it lead to new insights?I mean say in superconductivity would it lead to higher coherence--the field at a point x1,t1 is now causally connected with all x2,t2 at time-like separation from x1,t1?

If we take the non-relativistic limit of either the K.G. field equation or the Dirac field equation(where the field obeys commutator relations),we get the Schrodinger equation.Obviously at no point does the field turn into a wavefunction---so the Schrodinger equation also is an equation for the quantum field.The field obeys microscopic causality i.e. the field at x1,t1 does not commute with the field at x2,t2(time-like separated from x1,t1).Now my question is:--if we take this into account in phenomena of the condensed matter or in many body theory,would it lead to new insights?I mean say in superconductivity would it lead to higher coherence--the field at a point x1,t1 is now causally connected with all x2,t2 at time-like separation from x1,t1?

Er.... what is "higher coherence"?

How does "field at point x1,t1 is now causally connected with all x2,t2" would give a new insight into things like superconductivity? Is BCS theory not enough for conventional superconductivity as far as "insights" go?

I know this is vague,but my knowledge of superconductivity is quite limited.Could I say coherence length?

How does "field at point x1,t1 is now causally connected with all x2,t2" would give a new insight into things like superconductivity?

I was wondering if the fact that 'the field at x1,t1 does not commute with the field at x2,t2(time-like separated)' is taken into account in the phenomena of condensed matter?Does the many particle wavefunction that one writes down automatically satisfy the above?

I know this is vague,but my knowledge of superconductivity is quite limited.Could I say coherence length?

But the coherence length in a superconductor is tightly dependent on the coupling strength in a cooper pairs. And this depends very much on the exact mechanism of coupling, i.e. what is the glue that binds them together, how large is the bandwidth of the phonon spectrum, etc. You cannot get this simply writing the "Schrodinger equation".

I was wondering if the fact that 'the field at x1,t1 does not commute with the field at x2,t2(time-like separated)' is taken into account in the phenomena of condensed matter?Does the many particle wavefunction that one writes down automatically satisfy the above?

There are no "many-particle wavefunction". In practically all cases, you cannot write this. You can write the Hamiltonian, but you will seldom arrive at a "wavefunction". The many-body starting point for fermions, for example, is usually the single-particle spectral function which is the imaginary part of the single-particle Green's function. This ALREADY takes into account the many-body interaction via the self-energy term (both real and imaginary parts).

The whole point of many-body physics is the fact that you cannot write a wavefunction, at least not an accurate one, to describe the system. This is why you need "tricks", and intelligent tricks, to be able to handle such a system. This is the crux of many-body physics.

There are no "many-particle wavefunction". In practically all cases, you cannot write this. You can write the Hamiltonian, but you will seldom arrive at a "wavefunction". The many-body starting point for fermions, for example, is usually the single-particle spectral function which is the imaginary part of the single-particle Green's function. This ALREADY takes into account the many-body interaction via the self-energy term (both real and imaginary parts).

The whole point of many-body physics is the fact that you cannot write a wavefunction, at least not an accurate one, to describe the system. This is why you need "tricks", and intelligent tricks, to be able to handle such a system. This is the crux of many-body physics.

The picture that I had in mind was this:-there would be a Cooper pair field(where the Cooper pair would be a quantum of this field) and this field would obey microscopic causality or the field commutator relation.The Cooper pair field would form out of interaction of the electron field with the phonon field.But it seems this is not how things are done.

The picture that I had in mind was this:-there would be a Cooper pair field(where the Cooper pair would be a quantum of this field) and this field would obey microscopic causality or the field commutator relation.The Cooper pair field would form out of interaction of the electron field with the phonon field.But it seems this is not how things are done.

Then you will have a huge task ahead of you.

First of all, the "electron" that eventually forms the cooper pairs are not your typical, regular electron. They are quasiparticles, which means they themselves are single-particle excitation out of the many-body fermionic ground state. So already you are juggling with the many-body interaction to start with (we will not go into the cuprate superconductors that most say do not evolve out of well-defined quasiparticle states).

You then need to be able to show something similar to the Cooper problem where, upon addition of 2 additional quasiparticles just above the Fermi energy, the paring energy is more favorable and lower than this Fermi energy.

Before one even attempts that,one has to answer if there would be anything new added by a full-blown QFT approach and why.When you write down your Hamiltonian in terms of creation/destruction operators,field commuation relations are taken care of--equal time commutation relations are of course respected(what about commutation relation at two different space-time points--is that satisfied?).Are there other aspects of QFT which would be taken care of only in a full-blown QFT approach?

There are other questions I have in mind which I hope you can answer.As I said in post no. 1,there is now a Schrodinger equation for the field(from the consideration that Schrodinger equation is the non-rel. limit of K.G. or Dirac field equations) as well as for a wavefunction.What's the relation between the two?I tend to believe that there's nothing like a wavefunction of the particle--that was what Schrodinger did,that one needs to second quantize Schrodinger equation(the wavefunction then would be a functional of the field).What do you say?

Before one even attempts that,one has to answer if there would be anything new added by a full-blown QFT approach and why.When you write down your Hamiltonian in terms of creation/destruction operators,field commuation relations are taken care of--equal time commutation relations are of course respected(what about commutation relation at two different space-time points--is that satisfied?).Are there other aspects of QFT which would be taken care of only in a full-blown QFT approach?

I don't understand. What is a "full-blown QFT approach" as opposed to an "ordinary" QFT approach?

Superconductivity, superfluidity, quantum magnetism, etc. and in fact, ALL of condensed matter, make full use of the "full-blown QFT" approach. It's the only way to tackle such a problem since you literally cannot conceive of a "wave function" for such a system.

So aren't you really trying to figure out something that has already been done?

I've already indicated in post #5 what I mean by a full-blown QFT approach.I asked a few specific questions in post #7 which I am numbering below(kindly answer them one by one):-

(1.what about commutation relation at two different space-time points--is that satisfied?).2.Are there other aspects of QFT which would be taken care of only in a full-blown QFT approach?

There are other questions I have in mind which I hope you can answer.As I said in post no. 1,there is now a Schrodinger equation for the field(from the consideration that Schrodinger equation is the non-rel. limit of K.G. or Dirac field equations) as well as for a wavefunction.
3.What's the relation between the two?

4.I tend to believe that there's nothing like a wavefunction of the particle--that was what Schrodinger did,that one needs to second quantize Schrodinger equation(the wavefunction then would be a functional of the field).What do you say?

I've already indicated in post #5 what I mean by a full-blown QFT approach.I asked a few specific questions in post #7 which I am numbering below(kindly answer them one by one):-

I don't understand this. In Post #5, you said

The picture that I had in mind was this:-there would be a Cooper pair field(where the Cooper pair would be a quantum of this field) and this field would obey microscopic causality or the field commutator relation.The Cooper pair field would form out of interaction of the electron field with the phonon field.But it seems this is not how things are done.

This is a "full-blown QFT"? All I see is a conjecture in trying to make something like a a "cooper quantum field" (my terminology). Considering that the particles themselves (electrons, neutrinos, etc..) are NOT themselves a quantum field but rather ARISES out of the quantum field, aren't you jumping the gun a little bit? Shouldn't you FIRST show that your conjecture of making the particles themselves as the ground state field is valid?

And again, you did not address my point that these are NOT your regular electrons, but rather quasiparticles arising out of the Fermionic ground state (which themselve ARE the quantum field). So you are attempting to build another LAYER on top of an already existing field.

I don't see the utility of such thing. Is there a reason why you wish to do this? Is there something wrong with the field theoretic method of deriving the BCS theory as it is today?

Let's take a simpler example--a gas of electrons interacting via the Coulomb interaction.One way is to write down the Hamiltonian with the potential term taking care of the Coulomb interaction.Another way could be to treat electron-electron interactions in a QFT way(a la Feynman diagrams etc.).The latter is a full-blown QFT approach,the former is not.Now would you answer my questions 1 to 4 in my last post.

Let's take a simpler example--a gas of electrons interacting via the Coulomb interaction.One way is to write down the Hamiltonian with the potential term taking care of the Coulomb interaction.Another way could be to treat electron-electron interactions in a QFT way(a la Feynman diagrams etc.).

But what have you accomplished by doing that? It's a many-body problem that is unsolvable! What can you do with such a thing? Why do you think the Fermi Liquid theory exist?

The latter is a full-blown QFT approach,the former is not.Now would you answer my questions 1 to 4 in my last post.

I can't answer your questions 1 to 4. I'm not smart nor imaginative enough to do that, especially on something I don't understand.

This is a "full-blown QFT"? All I see is a conjecture in trying to make something like a a "cooper quantum field" (my terminology).

Dump the cooper field.You have electrons(regular) and phonons--electrons interact with each other via the Coulomb interaction as well as via the lattice vibrations.If one 'could' treat this in a full QFT manner,would it add value or is expected to add more value than BCS.I am not proposing that one do it that way--I am just wondering/asking if there is any feature of QFT that's possibly not taken care of in the present approaches to many body problems in cond. matt.A specific question is regarding the field commutator at two different spacetime points.My guess is that it is automatically satisfied when you write down things(Hamiltonian) in terms of creation/destruction operators.May be,I need to dig out my old notes or sit down and work it out,to be sure.There are other questions as well(2-4) which need to be answered.

Dump the cooper field.You have electrons(regular) and phonons--electrons interact with each other via the Coulomb interaction as well as via the lattice vibrations.If one 'could' treat this in a full QFT manner,would it add value or is expected to add more value than BCS.I am not proposing that one do it that way--I am just wondering/asking if there is any feature of QFT that's possibly not taken care of in the present approaches to many body problems in cond. matt.

I would pay money to see you do this without invoking any mean-field approximation. Remember, you have the order of Avogadro's number of interaction, PLUS the phonon FIELD that is already a product of QFT treatment.

I still want to know what is so not-QFT about the BCS theory derivation and Fermi Liquid theory. The way you are describing things done in condensed matter sounds as if we don't use field theoretic methods at all!

If we take the non-relativistic limit of either the K.G. field equation or the Dirac field equation(where the field obeys commutator relations),we get the Schrodinger equation.Obviously at no point does the field turn into a wavefunction---

The solutions to both the KG equation and the Dirac equation are wavefunctions.

so the Schrodinger equation also is an equation for the quantum field.

Indeed. It is explained in Zee's Quantum Field Theory in a Nutshell on pp.84-85 that quantum mechanics is a (0+1) dimensional quantum field theory. What we are normally accustomed to calling "quantum field theory" is the (3+1) dimensional generalization of quantum mechanics.

I would pay money to see you do this without invoking any mean-field approximation. Remember, you have the order of Avogadro's number of interaction, PLUS the phonon FIELD that is already a product of QFT treatment.

Of course,something like mean field theory needs to be used,but in one case(the one that's used) you take the mean of Coulomb interactions(of an electron with other electrons,considering an electron gas) which you take to be dependent on the 'instantaneous' positions of the electrons, and in another(QFT) you would have to consider interaction of an electron with other electrons via photon exchange and calculate its mean effect.The latter may be difficult to calculate,but if it can be done it would be more accurate and probably throw more light on the phenomena of cond. matt.

Tom,if you are to do QFT in a non-rel. scenario(i.e. in problems of cond. matt.),why not go the whole way and do it properly.What prevents you from second-quantizing the Schrodinger equation itself and then applying it to problems of cond. matt.?I don't know if using schemes like the slater determinant, or writing the hamiltonian in terms of creation/destruction operators takes into account all features of QFT.

Of course,something like mean field theory needs to be used,but in one case(the one that's used) you take the mean of Coulomb interactions(of an electron with other electrons,considering an electron gas) which you take to be dependent on the 'instantaneous' positions of the electrons, and in another(QFT) you would have to consider interaction of an electron with other electrons via photon exchange and calculate its mean effect.The latter may be difficult to calculate,but if it can be done it would be more accurate and probably throw more light on the phenomena of cond. matt.

Er... May I suggest you look at a book such as Mahan's "Many-Particle Physics", or Mattuck's "Feynman Diagrams in Many-Body Problem"? You'd be surprised what has ALREADY been done. This is especially important when it appears that you think the mean field approximation hasn't been done via QFT. I also strongly suggest you read up on Landau's Fermi Liquid theory.

Tom,if you are to do QFT in a non-rel. scenario(i.e. in problems of cond. matt.),why not go the whole way and do it properly.

You can indeed use a nonrelativistic QFT. I have a book on exactly that by Henley and Thirring. But you didn't mention nonrelativistic QFT in the post I quoted, you mentioned the KG and Dirac equations, saying that their solutions don't "become" wavefunctions in the nonrelativistic limit. My point is that they already are wavefunctions.

You can indeed use a nonrelativistic QFT. I have a book on exactly that by Henley and Thirring. But you didn't mention nonrelativistic QFT in the post I quoted, you mentioned the KG and Dirac equations, saying that their solutions don't "become" wavefunctions in the nonrelativistic limit. My point is that they already are wavefunctions.

If you begin with a K.G. equation or Dirac equation for the wavefunction and take the non-relativistic limt,you get Schrodinger equation for the wavefunction.But if you begin with K.G./Dirac field equation(where the field obeys commutation relations) and take the non-rel. limit,you'll get Schrodinger equation for the field.

Now I want to go a step further and say that just as K.G./Dirac equation for the wavefunction describes quite a few things very well,it's not the final answer and that one needs to second quantize it,the non-rel. limit of K.G./Dirac (wavefunction) equation namely Schrodinger (wavefunction) equation is only an approximate description of reality and one needs to second quantize it.

In fact,I think the notion of the wavefunction of a particle is a wrong notion.From QFT we know that the wavefunction is a functional of the field--similarly in non-rel. QM,the wavefunction is a functional of the field where the field obeys Schrodinger equation.Historically,Schrodinger equation was discovered as an equation for a particle,it wasen't second quantized and the notion of wavefunction of a particle arose.I may be wrong,but I think the notion of Schrodinger equation as an equation for the wavefunction(of a particle) is wrong--interpretation of the Schrodinger equation as an equation for the field(where the field obeys certain commutation relations)seems correct to me.